Trigonometry A

Average Error: 0.1 → 0.1

Time: 4.3s

Precision: 64

\[0.0 \le e \le 1\]

Math

\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

↓

\[\left(e \cdot \sin v\right) \cdot \frac{1}{1 + e \cdot \cos v}\]

C

double f(double e, double v) {
        double r8591 = e;
        double r8592 = v;
        double r8593 = sin(r8592);
        double r8594 = r8591 * r8593;
        double r8595 = 1.0;
        double r8596 = cos(r8592);
        double r8597 = r8591 * r8596;
        double r8598 = r8595 + r8597;
        double r8599 = r8594 / r8598;
        return r8599;
}

↓

double f(double e, double v) {
        double r8600 = e;
        double r8601 = v;
        double r8602 = sin(r8601);
        double r8603 = r8600 * r8602;
        double r8604 = 1.0;
        double r8605 = 1.0;
        double r8606 = cos(r8601);
        double r8607 = r8600 * r8606;
        double r8608 = r8605 + r8607;
        double r8609 = r8604 / r8608;
        double r8610 = r8603 * r8609;
        return r8610;
}

Error

Bits error versus e

Bits error versus v

Derivation

1. Initial program 0.1

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
2. Using strategy rm

3. Applied div-inv 0.1

   \[\leadsto \color{blue}{\left(e \cdot \sin v\right) \cdot \frac{1}{1 + e \cdot \cos v}}\]

4. Final simplification 0.1

   \[\leadsto \left(e \cdot \sin v\right) \cdot \frac{1}{1 + e \cdot \cos v}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))